
which implies that \(\tilde{U}\left(x\right)\)  is \(\left(\frac{\mu }{2},\theta \right)\) -degenerated
convex outside the ball. Now, we construct \(\hat{U}(\ x)\)  so that
it is twice differentiable, degenerated convex on all \(\mathbb {R}^{d}\) 
and differs from \(U(\ x)\)  less than \(4LR^{1+\alpha }+4LR^{1+\ell +\alpha }+\frac{4\mu }{\left(2-\theta \right)}\ R^{2-\theta }\) .
Based on the same construction of [1]}, we first
define the function \(V\)  as the convex extension [2]}
of \(\tilde{U}\)  from domain \(\Omega =R^{d}\setminus \mathbb {B}\left(0,R\right)\) 
to its convex hull \(\Omega ^{co}\) , \(V\left(x\right)=\inf \left\lbrace  \sum _{i}\lambda _{i}\tilde{U}(\ x_{i})\right\rbrace  \) 
for every \(x\in \mathbb {R}^{d}.\)  Since \(\tilde{U}(\ x)\)  is convex
in \(\Omega \) , \(V(\ x)=\tilde{U}(\ x)\)  for \(\ x\in \Omega \) . By Lemma REF ,
\(V(\ x)\)  is convex on the entire domain \(R^{d}\)  and \(V(\ x)\)  can
be represented as
\(V(\ x)=\inf _{\begin{array}{c}\lbrace \ x_{j}\rbrace \subset \Omega ,\\\left\lbrace  \lambda _{j}\big |\sum _{j}\lambda _{j}=1\right\rbrace  \\\text{s.t.},\sum _{j}\lambda _{j}\ x_{j}=\ x,\mathrm {and}\left\Vert x_{i}\right\Vert =R\end{array}}\left\lbrace  \sum _{j}\lambda _{j}\tilde{U}(\ x_{j})\right\rbrace  .\) 
